Question : If $a + b + c = m$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$ , then the average of $a^{2}, b^{2},$ and $c^{2}$ is:
Option 1: $m^{2}$
Option 2: $\frac{m^{2}}{3}$
Option 3: $\frac{m^{2}}{9}$
Option 4: $\frac{m^{2}}{27}$

Question

Question : If $a + b + c = 0$ , then the value of $\frac{a^{2}}{b c} + \frac{b^{2}}{c a} + \frac{c^{2}}{a b}$ is:
Option 1: 1
Option 2: 3
Option 3: - 1
Option 4: 0

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Answer 1

Correct Answer: $\frac{m^{2}}{3}$

Solution : Given: $a + b + c = m$
Squaring both sides, we get,
⇒ $(a + b + c)^{2} = m^{2}$
⇒ $a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = m^{2}$ ------(1) [Using $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$ ]
$∵ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$
⇒ $\frac{b c + a c + a b}{a b c} = 0$
⇒ $b c + a c + a b = 0$
Putting the value of $b c + a c + a b = 0$ in equation (1), we have,
⇒ $a^{2} + b^{2} + c^{2} + 2 (0) = m^{2}$
⇒ $a^{2} + b^{2} + c^{2} = m^{2}$
Dividing by 3, both sides to find the average,
⇒ $\frac{a^{2} + b^{2} + c^{2}}{3} = \frac{m^{2}}{3}$
Hence, the correct answer is $\frac{m^{2}}{3}$ .

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Question : If $a + b + c = m$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$ , then the average of $a^{2}, b^{2},$ and $c^{2}$ is:
Option 1: $m^{2}$
Option 2: $\frac{m^{2}}{3}$
Option 3: $\frac{m^{2}}{9}$
Option 4: $\frac{m^{2}}{27}$